Theorem 32.4 from Munkres' Topology book:
Every well-ordered set $X$ is normal in the order topology
This is something a continuation of this post, where, evidently, I forgot what the definition of a well-ordered space. In this post, I am trying to prove the following claim Munkres uses in his proof of theorem 32.4:
Let $B$ be closed in $X$, and $a \in X-B$. Then there exists a basis element about $a$ which is disjoint from $B$ and contains $(x,a]$ for some $x \in X$.
I have tried to prove this, but I don't know how to. I could use some help.